Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations

Abstract

In this paper, we shall study the almost sure pathwise exponential stability property for a class of stochastic functional differential equations with delays, possibly, in the highest-order derivative terms driven by multiplicative noise. Instead of establishing a moment exponential stability as the first step and then proceeding to investigate the pathwise stability of the system under consideration, we shall develop a direct approach for this problem. As a consequence, we can show that some systems, which are not exponential momently stable, have the exponential stability not sensitive to small delays in the almost sure sense.

where \(y(0) = y_0\in {\mathbb {R}}\), and \(\alpha \), \(\beta \in {\mathbb {R}}\). Since \(y(t) = y_0 e^{(\alpha +\beta )t}\) for all \(t\ge 0\), it is easy to see that if \(\alpha +\beta <0\), the null solution of (1.1) is exponentially stable.

Because of the time delay feature in (1.2), one need set up proper initial data different from those in (1.1) to make (1.2) well-defined. For instance, let \(y(0)=\phi _0\in {\mathbb {R}}\) and \(y(t)=\phi _1(t)\) for \(t\in [-r, 0]\) where \(\phi _1\in L^2([-r, 0], {\mathbb {R}})\). It is well known (see, e.g., [2] or [14]) that under the condition \(\alpha +\beta <0\), there exists \(r_0>0\) such that the null solution of (1.2) is exponentially stable for all \(r\in (0, r_0)\), while for \(r>r_0\), the null solution is exponentially unstable. In other words, exponential stability at this moment is not sensitive to small delays, a result which actually remains true in any finite dimensional context (see, e.g., Chap. 7 in [2]).

where \(r>0\). For pathwise stability of Eq. (1.6), it turns out that, in addition to the condition \(\alpha +\beta < \sigma ^2/2\), the null solution of Eq. (1.6) could secure the same exponential stability in the almost sure sense as long as the delay parameter \(r>0\) is sufficiently small. In fact, it is shown in Bierkens [3] that if \(r>0\) is so small that

then the null solution of (1.6) is exponentially stable almost surely. Thus, stability is not sensitive in this case to small delays. It is also worth mentioning that as a corollary of more general results for nonlinear stochastic systems, Appleby and Mao [1] showed that if \(r>0\) is such a small number that

then the null solution of (1.6) has the same pathwise exponential stability. Here \(\Phi \) is the cumulative distribution function of a standard normal random variable.

In an infinite dimensional setting, the situation is quite different. In particular, it was observed by Datko et al. [6] (see also [5]) that small delays may destroy stability for a partial differential equation. More precisely, consider a deterministic differential equation in a Hilbert space H,

where A generates a \(C_0\)-semigroup and B is a linear operator in H. If the spectrum \(\sigma (A)\) of A is unbounded along an imaginary line, it was shown (see Theorem 7.4 in [2]) that for any \(r_0>0\), one can always find an operator \(B\in {\mathscr {L}}(H)\), the family of all bounded linear operators on H, and \(r\in (0, r_0)\) such that \(A+B\) generates an exponentially stable semigroup and meanwhile the system (1.7) is not exponentially stable. From this observation, we can see that the unboundedness of the spectrum set of A along imaginary axes may cause trouble in the stability analysis of (1.7). Therefore, one need make additional assumptions on the semigroup generated by A. Note that in finite dimensional spaces, the spectrum set of A is always bounded. In fact, we have the following result which clearly implies the fact that for finite dimensional equations, the exponential stability cannot be destroyed by small delays.

Theorem 1.1

Assume that A generates a norm continuous \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), i.e., \(e^{\cdot A}:\, [0, \infty )\rightarrow {\mathscr {L}}(H)\) is continuous, and the semigroup generated by \(A+B\) is exponentially stable in H. Then, there exists a constant \(r_0>0\) such that the system (1.7) is exponentially stable for all \(r\in (0, r_0)\).

Proof

In this work, we want to consider the sensitivity problem of pathwise stability to small delays for stochastic retarded evolution equations in Hilbert spaces. Based on the above statements, especially Theorem 1.1, let us first consider a concrete example of linear stochastic partial differential equations to motivate our theory,

where \((\phi _0, \phi _1)\) is some properly given initial datum. Here, the problem we want to address is whether, under the condition \(\alpha _0 -1<\sigma ^2/2\), the null solution of Eq. (1.10) secures the same pathwise exponential stability, at least for sufficiently small delay parameter \(r>0\).

Note that this problem cannot be addressed by the usual method dealing with pathwise exponential stability, i.e., first considering moment stability and then establishing pathwise exponential stability. To illustrate this, let us consider a simple example of one-dimensional stochastic differential equation,

Thus, the null solution is mean square exponentially stable if and only if \(\beta <-\sigma ^2/2\), a condition which is much more restrictive than \(\beta <\sigma ^2/2\).

In this work, we shall employ a different method to address the pathwise sensitivity problem to small delays. In particular, as a by-product of the general theory established later on, we shall answer this question affirmatively for (1.10).

where Q is a positive, self-adjoint and trace class operator, \(Tr(Q)<\infty \), on K. In particular, we shall call such \(W_Q(t)\), \(t\ge 0\), a K-valued Q-Wiener process with respect to \(\{{\mathscr {F}}_{t}\}_{t\ge 0}\). We introduce the subspace \(K_Q=\hbox {Ran}\,Q^{1/2}\), the range of \(Q^{1/2}\), of K and let \({\mathscr {L}}_2={\mathscr {L}}_2(K_Q, H)\) denote the space of all Hilbert–Schmidt operators from \(K_Q\) into H.

Since \({\mathscr {D}}(A)\) is dense in H, the relation (2.7) follows easily from (2.9). Last, since the semigroup \(e^{tA}\), \(t\ge 0\), is exponentially stable and P is the unique solution of (2.4), one can see that the family of nonnegative, self-adjoint operators \(P(t)\equiv P\in {\mathscr {L}}(H)\) is the unique solution of (2.7) with \(t=0\), \(T= +\infty \) and \(G=P\). Therefore, P satisfies (2.6) with \(G=P\) which is exactly the equality (2.2). Thus, the proof is complete. \(\square \)

Lemma 2.2

Suppose that A generates a \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), such that

Proof

Thus, the exponential stability of \(e^{tA}\) implies the same property of \(e^{tA_n}\) for \(n\in \rho (A)\). By virtue of Lemma 2.1, for any self-adjoint, nonnegative operator \(L\in {\mathscr {L}}(H)\), there exists a unique solution \(P_n\in {\mathscr {L}}(H)\) to (2.10) for \(n\in \rho (A)\).

We show that for all \(x,\,y\in H\) and \(j\in {\mathbb {N}}\), there is the relation \(\langle x, (P_n(j)-P(j)) y\rangle _H\rightarrow 0\) as \(n\rightarrow \infty \). Indeed, we can see this by induction. For \(j=0\), the claim holds trivially. Suppose now that the claim holds for \(j=k-1\), \(k\ge 1\). Let \(x,\,y\in H\), and then for \(j=k\),

by virtue of Dominated Convergence Theorem. Hence, the claim is proved by induction.

Last, by the proofs of Lemma 2.1 we have that \(P(j)\rightarrow P\) and \(P_n(j)\rightarrow P_n\), uniformly in n, as \(j\rightarrow \infty \) in the norm topology of \({\mathscr {L}}(H)\). Therefore, we obtain that

where \(B_i\in {\mathscr {L}}(H)\) and \(w_i(t)\), \(t\ge 0\), \(i=1,\,\ldots ,\, n\), are a group of independent, standard real-valued Brownian motions. To proceed further, we first recall two useful lemmas which are important in their own right. The first lemma is a strong law of large numbers for continuous local martingales (see, e.g., [9] or Theorem 1.3.4 in [12]), and the proof of the second is referred to that of Proposition 2.1.4 in [10].

Lemma 2.3

Let M(t), \(t\ge 0\), be a real-valued, continuous local martingale with \(M(0)=0\). If its quadratic variation [M] satisfies that

For each \(m\in {\mathbb {N}}\), let \(\Lambda _m :=A_m +\beta \cdot B + \frac{\Vert \beta \Vert ^2}{4} +\gamma \) where \(A_m\) is the Yosida approximation of A. Without loss of generality, it is assumed that for all \(m\ge 1\), we have

Since semigroups \(T(\cdot )\) and \(T_m(\cdot )\), \(m\ge 1\), are exponentially stable, by virtue of (2.17) and Lemma 2.1, one can find for \(m\in {\mathbb {N}}\) and \(\Delta (I) \in {\mathscr {L}}^+(H)\) a unique solution \(Q_m\in {\mathscr {L}}(H)\) to the following Lyapunov equations

If \(y_0=0\), then \({\mathbb {P}}(y(t, 0)=0)=1\) for \(t\ge 0\), and the desired estimate holds trivially. Now suppose that \(y_0\not = 0\), then by uniqueness of the solutions and strict positivity of \(P_m\), \({\mathbb {P}}(\langle P_my_m(t), y_m(t)\rangle _H=0)=0\) for all \(t\ge 0\) (or, the reader is referred to a proof similar to that of Lemma 2.1 in [11]). Letting \(\theta _m(t) = \langle P_my_m(t), y_m(t)\rangle _H\), \(t\ge 0\), and applying Itô’s formula to \(\log \theta _m(t)\), we obtain by (2.22) that for \(t\ge 0\),

However, this is immediate since we can take \(\beta = -2\sigma \) and \(\lambda = 3\sigma ^2/2 +\varepsilon \) with \(\varepsilon >0\) small enough by using the condition \(\alpha < \sigma ^2/2\) and the mild solution y is pathwise exponentially stable with

3 Stochastic Delay Evolution Equations

Let H be a real Hilbert space and V be another Hilbert space such that V is dense in H and the inclusion map \(V\hookrightarrow H\) is continuous. The norms and inner products of H, V are denoted by \(\Vert \cdot \Vert _H\), \(\Vert \cdot \Vert _V\) and \(\langle \cdot , \cdot \rangle _H\), \(\langle \cdot , \cdot \rangle _V\), respectively. By identifying the dual of H with H, we may assume that

where \(W^{1, 2}([0, T]; V^*)\) is the classic Sobolev space consisting of all functions \(y\in L^2([0, T]; V^*)\) such that y and its first-order distributional derivatives are in \(L^2([0, T]; V^*)\) and C([0, T]; H) is the space of all continuous functions from [0, T] into H, respectively. The duality pair between V and \(V^*\) is denoted by \(\langle \!\langle \cdot , \cdot \rangle \!\rangle _{V, V^*}\). Let \(a:\, V\times V\rightarrow {\mathbb {R}}\) be a bounded bilinear form satisfying Gårding’s inequality

Then, \(A\in {\mathscr {L}}(V, V^*)\), the family of all bounded and linear operators from V to \(V^*\). The realization of A in H, which is the restriction of A to the domain \({\mathscr {D}}(A) =\{x\in V:\, Ax\in H\}\), is also denoted by A. Hence, \(\langle x, Ay\rangle _H = \langle \!\langle x, Ay\rangle \!\rangle _{V, V^*}\) for all \(x\in V\) and \(y\in V\) with \(Ay\in H\). It is well known that A generates an analytic \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), in \(V^*\) such that \(e^{tA}:\, V^*\rightarrow V\) for each \(t>0\). In this work, we always suppose, for simplicity, that \(\lambda =0\) in (3.2) and A generates an exponentially stable \(C_0\)-semigroup with \(\Vert e^{tA}\Vert \le e^{-\alpha t}\) for all \(t\ge 0\).

where \(A_1\in {\mathscr {L}}(V, V^*)\) and \(A_2(\cdot )\in L^2([-r, 0]; {\mathscr {L}}(V, V^*))\). In particular, a function \(y\in L^2([-r, T]; V)\cap W^{1, 2}([0, T]; V^*)\) which satisfies (3.3) is called a strong solution of (3.3) in \([-r, T]\). It has been shown (see [7] or [8]) that there exists a unique solution y of (3.3) such that

Then, it has been shown (see [7]) that \({\mathcal {T}}(t)\), \(t\ge 0\), is actually a \(C_0\)-semigroup on \({\mathcal {H}}\). Let \({\mathcal {A}}\) be the infinitesimal generator of \({\mathcal {T}}(t)\) or \(e^{t{\mathcal {A}}}\), \(t\ge 0\), and then \({\mathcal {A}}\) is completely described by the following theorem.

Note that \(C_0\)-semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), allows us to transfer the exponential stability problem of the time delay system (3.3) to that one of a non-time delay system. To see this, we can rewrite (3.3) as a Cauchy problem,

where \(Y(t)=(y(t), y(t+\cdot ))\) is the lift-up function of y(t), \(t\ge 0\). To show the exponential stability of the null solution of (3.3) in H, it suffices to study the corresponding exponential stability of system (3.7). Indeed, assume that the null solution of (3.7) is exponentially stable, and then there exist constants \(M>0\) and \(\mu >0\) such that

Hence, it is natural to consider exponential stability of the non-time delay system (3.7) rather than time delay one (3.3). To establish the exponential stability of the \(C_0\)-semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), one may want to develop a dissipative operator program by using Lemma 2.4. However, the following example shows some difficulties in association with this scheme.

If \(\beta =0\), the solution \(u(t)= e^{-\alpha t}\), \(t\ge 0\), which is exponentially stable. Otherwise, it is clear that \({\mathcal {H}}= {\mathbb {R}}\times L^2([-1, 0]; {\mathbb {R}})\) equipped with the usual inner product. If we take the above viewpoint to consider system (3.7), then it turns out in this case that for any number \(a>0\) and \(\phi \in {\mathscr {D}}({\mathcal {A}})\),

In other words, it is impossible, according to Lemma 2.4, to find a constant \(a>0\) such that \(\Vert e^{t{\mathcal {A}}}\Vert \le e^{-a t}\), \(t\ge 0\) in spite of the fact that \(\Vert e^{tA}\Vert \le e^{-\alpha t}\), \(t\ge 0\). To avoid this difficulty, we can introduce an equivalent inner product \((\cdot , \cdot )_{\mathcal {H}}\) or norm \(|\cdot |\) to the canonical one \(\langle \cdot , \cdot \rangle _{\mathcal {H}}\) or \(\Vert \cdot \Vert \) on \({\mathcal {H}}\) so as that the semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), becomes exponentially stable under the norm \(\Vert \cdot \Vert \).

Proof

We show that there exists an equivalent inner product \((\cdot , \cdot )_{\mathcal {H}}\) on \({\mathcal {H}}= H\times L^2([-r, 0]; V)\) to the canonical one \(\langle \cdot , \cdot \rangle _{\mathcal {H}}\) on \({\mathcal {H}}\) such that

which implies that the inner product \((\cdot , \cdot )_{\mathcal {H}}\) is also equivalent to the canonical inner product \(\langle \cdot , \cdot \rangle _{\mathcal {H}}\) on \({\mathcal {H}}\). On the other hand,

Hence, by virtue of Proposition 2.1 and Theorem 3.2, we have that if there exists a number \(\beta \in {\mathbb {R}}\) such that \({A}+\beta {B}\) generates an exponentially stable \(C_0\)-semigroup and

Last, let us study an example in which the delay appears in the highest-order derivative term of the stochastic system to close this work. The example (1.10) in Section 1 can be also treated in a similar manner.

Example 4.1

First consider the following linear stochastic partial differential equation,

then the null solution of (4.7) is exponentially stable in the almost sure sense. It may be verified that \(\beta =-2\sigma \) and \(\lambda = \frac{3}{2}\sigma ^2 +\varepsilon \) with \(\varepsilon >0\) sufficiently small satisfy condition (4.8). In this case, the third inequality in (4.8) is actually reduced to

In other words, in addition to the condition \(-1 -\alpha _0<\sigma ^2/2\) which ensures the pathwise exponential stability of Eq. (4.6), if we further assume that the relation (4.9) is true, then the null solution of (4.7) is exponentially stable in almost sure sense. Thus, the stability is not sensitive to small delays.

Copyright information

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

This article is published under an open access license.
Please check the 'Copyright Information' section for details of this license and
what re-use is permitted.
If your intended use exceeds what is permitted by the license or if
you are unable to locate the licence and re-use information,
please contact the Rights and Permissions team.